So long as a body moves freely no force is appreciated by it. A falling aviator (neglecting air resistance) will not appreciate any gravitational force. He follows a natural track, thereby freeing himself from the force experienced in contact with matter. He acquires an accelerating motion with respect to an inertial system. By acquiring a particular accelerating motion an observer can annul any force experienced in any small region where the field of force can be considered constant.
Thus Einstein, interpreting the equality of gravitational and inertial mass, showed that the same quality manifests itself according to circumstances as “weight” or as inertia, and that all force is purely relative and may be treated as one phenomenon (an interruption in energy flow). This “Principle of Equivalence” shows that small portions of the World-Fabric, observed from a freely moving particle (free of force), could be treated as small portions of the World-Frame.[3]
If such observations were practicable, we could determine the Fabric curvature by referring point-event measurements to equation (1). We cannot observe from unique tracks but we can observe them from our restrained situation. Their importance is now apparent, because, by tracing them over a region, we are tracing something absolute in the Fabric—its geometrical character. We study this curvature by exploring separation-intervals on the tracks of freely moving bodies, relating these separation-intervals to actual measurements in terms of space and time components depending on the observer’s reference system. The law of curvature must be the law of gravitation. To illustrate the lines on which Einstein proceeded to survey the World-Fabric from the earth we will consider a similar but more simple problem—the survey of the sea-surface curvature from an airship. We study this curvature by exploring small distances on the tracks of ships (which we must suppose can only move uniformly on unique tracks—arcs of great circles), relating such distances to actual measurements in terms of length and breadth components depending on the observer’s reference system. This two-dimensional surface problem can be extended to the four-dimensional Fabric one.
We consider the surface to be covered by two arbitrarily drawn intersecting series of curves: curves in one series not intersecting each other, vide figure. This Gaussian system of coordinates is appropriate only when the smaller the surface considered, the more nearly it approximates to Euclidean conditions. It admits of defining any point on the surface by two numbers indicating the curves intersecting at that point. P is defined by
,
.
